Nodal Admittance Matrix is built to model or represent the electrical system behaviour. In such text book authored by Grainger has given a clear picture to derive the equation and creating the matrix. The matrix is built by represent the voltage drop across a branch into its nodal voltage. After this representation,we build matrix each branch using building block and combine together all of the building block to form a nodal admittance matrix. I think that a simple thinking of all of the matter. Let us see the example netwrok from Grainger:

this is the matrix which is produced from combining the building block matrix at each branch:

If we see the Nodal Admittance Matrix above, we can see that the matrix shows an elegant and simple shape of the Kirchoff equation of current summation in a node. For the matrix above, there will be four variables with four equations. Each equation is representing Current Kirchoff Law in each node. This equation can be seen clearly if we multiply each row. For example: The multiplication the first row of the nodal admittance matrix with voltage matrix will show a Current Kirchoff summation at node 1. In another word we can build this matrix directly from the kirchooff equation. Just stack on the naming convention we have made to each node and create the matrix assosiating with its nodes. But, i think this way is more difficult 😀

This is the reason why the Nodal Admittance Matrix represent the system behaviour. If we solve the equation above, we will get voltage at each node and finally will know the load flow of the network. In further modification of the matrix, reducing the matrix for example, it means that we are doing a Network Reduction.